A circle's center is at #(4 ,2 )# and it passes through #(6 ,7 )#. What is the length of an arc covering #(5pi ) /3 # radians on the circle?

1 Answer
Tony B
Apr 6, 2016

Arc length#~~28.2# to 1 decimal place

Explanation:

Let the radius of the circle be #r#
Let the length of arc be #L_a#

Distance from the circles centre to any point on its circumference is always the same.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine the radius of the circle")#

#=> r=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

#=> r = sqrt((6-4)^2 +(7-2)^2)#

#=>color(blue)(r = sqrt(29))" "-># 29 is a prime number so can not be simplified

'~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine the length of arc")#

#color(brown)("Important point:")#

#color(brown)("1 radian is such that its length of ark is the same as")# #color(brown)("the length of the radius.")#

So the length of arc #L_a=rxx (5pi)/3#

#=> L_a=sqrt(29)xx (5pi)/3#

#color(blue)(L_a~~28.2" to 1 decimal place")#

'~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Check:")#

Circumference = #piD = pixx2sqrt(29)#

and we have #1 2/3" of "1/2 # of the circumference

#=>L_a=1 2/3 xxpisqrt(29) =28.19...# Confirmed!

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